Inflow conditions with realistic turbulent features are generated from an initially laminar flow by adding random perturbations in a precursor simulation with cyclic horizontal boundary conditions. To study the potential of DART under different inflow conditions, eight boundary layers (BLs) ranging from a neutral to a strongly stable BL are used as reference inflow conditions, all having approximately the same wind direction and wind speed at hub height. As reported by , wake steering is ineffective in a convective boundary layer, which is therefore not considered in this study. Due to the large computational expense it was not possible to increase the number of simulations. Although these eight BLs do not capture the great variability in the free field, it is considered sufficient to provide a proof of concept for data-driven models.

The total domain size and simulation time vary between the BLs, are determined empirically until convergence to a stationary state occurs and are dependent on the size of the largest eddies that explicitly need to be resolved. The details of the precursor simulations are summarized in Table .

BL1 and BL2 portray neutral conditions with roughness lengths representing low crops (z0= 0.1 m) and parkland (z0= 0.5 m), which are typical landscapes found in northern Germany. Following , constant cooling rates at the surface are prescribed to generate stable BLs, where BL3 and BL4 represent weakly stable (∂θ∂t-1=-0.125Kh-1) and BL5 and BL6 strongly stable conditions (∂θ∂t-1=-0.25Kh-1; following ). Two additional (near-)neutral BLs are generated, one representing grassland (z0= 0.03 m) and one having a very small negative sensible heat flux (H= -0.001 Kms-1), which is in the acceptable range defined in .

Stationary inflow conditions are taken at 2.5 rotor diameters (D) upstream of the turbines simulated in the main simulations (Sect. ) and averaged over a line of size 2 D in the crosswise direction and a period of 60 min. These inflow conditions are assumed to be undisturbed, hence far enough from the turbine that induction does not play a role. Typical inflow parameters are displayed in Fig. , indicating that the wind speed is comparable for all simulations but that the atmospheric conditions vary. A more stable boundary layer, indicated by a larger Obukhov stability parameter (zL-1), typically has a higher shear (α) and veer (∂α) and lower turbulence intensity (TI). The spread of the parameters between the main simulations (see Sect. ) in the same boundary layer, indicated by the standard deviation as whiskers in Fig. , is small enough to be neglected.

After generating stationary inflow conditions with a precursor, simulations with one turbine are performed. Information on turbulence characteristics from the precursor simulation is fed to the main simulation by adding a turbulent signal to a fixed mean inflow (turbulence recycling method) far upstream of the turbine. A radiation boundary condition ensures undisturbed outflow downstream of the simulated turbine. The size of the recycling area is equal to the domain size of the precursor simulation, and the domain size of the main simulation is only extended in streamwise direction by placing a turbine at x= 6 D downstream of the recycling area. Wake data until x= 10 D are used for analysis, but the domain is extended to x= 13 D to eliminate blockage effects. The total simulation time is 80 min: the first 20 min are considered spin-up time, and the last 60 min are used for analysis.

The simulated turbine is an actuator disc model with rotation (ADMR) representing a 5 MW NREL turbine, having a hub height of 90 m and a rotor diameter D of 126 m , as implemented in . Turbine yaw angles (ϕ) of -30, -15, 0, 15 and 30∘ are simulated, where a positive yaw angle is defined here as a clockwise rotation of the turbine when looking from above. Pitch angles (β) of 0, 2.5 and 5∘ are simulated to study the effect of the thrust force on downstream wake characteristics. This adds up to a total of 120 main simulations with one turbine, i.e., 15 turbine settings (5 yaw angles times 3 pitch angles) for each of the 8 inflow conditions. The effect of ϕ and β on the thrust coefficient CT is illustrated in Fig. , illustrating that the effect of the turbine yaw angle of the thrust coefficient is approximately symmetric around zero.

It should be noted that smaller step sizes for yaw and pitch angles would be preferred as these step sizes could be too coarse when utilizing a regression-based model (Sect. ). This can lead to deviating estimates when interpolating to values far away from these set points (e.g., for ϕ= 7.5∘). Increasing the step size would, however, lead to more simulations, which was computationally not feasible and not deemed necessary to show proof of concept.

The wake is described using the normalized wake deficit, defined as und=uwake-u∞u∞,h, where uwake represents the observed wind speed in the wake, u∞ the undisturbed inflow 2.5 D upstream at the same height and u∞,h the undisturbed inflow at hub height. It is assumed that the advection velocity is constant in streamwise direction (assumption of frozen turbulence) and that the wake behaves as a passive tracer in the ambient wind .

A data-driven model will not be able to produce a full multidimensional flow field but rather estimate parameters describing the wake at desired downstream positions. Since curled wakes are considered, key wake steering parameters in this study are retrieved with the multiple 1D Gaussian method . In the example below, the wake of a turbine with a +30∘ yaw angle in BL1 at x= 5 D is considered (Fig. a). This method fits a simple 1D Gaussian at every vertical level (k=1…K) where information is available to obtain a set of local normalized wake center deficits (A=A1…AK), wake center positions (μ=μ1…μK) and wake widths (σ=σ1…σK). Subsequently, another Gaussian can be fitted through the local wake center deficits in the vertical (Fig. b) to find the overall normalized wake center deficit (Az) and vertical position with respect to hub height (μz), as well as the vertical extension of the wake (σz). The local wake center position and width at vertical level k that corresponds to μz are subsequently considered to be lateral wake center position (μy) relative to the turbine location and wake width (σy). Next, by fitting a second-order polynomial through the local wake center positions between upper- and lower-tip height (Fig. c), one obtains a measure for the curl (coefficient of quadratic term) and tilt (coefficient of linear term) of the wake. An expression for the wake width as a function of height is found by repeating this step for the local wake widths (Fig. d) to obtain coefficients sa and sb. After this procedure, the wake can be described by the set of dimensionless parameters displayed in Table .

Note that this method cannot accurately capture the splitting of the wake in two separate cells, which might occur under strong veer as discussed in . Such cases will result in inaccurate values for the key wake steering parameters and should be filtered out before applying the regression model described in Sect. .

A regression model (Sect. ) is used to estimate the key wake steering parameters in Table . A set of measurable inflow and turbine variables are used as input parameters, which are made dimensionless to make the model more universally applicable, at least within the variability found between the simulations in this study. This set of parameters is presented in Table .

Although these input parameters might all have their own isolated effect on the wake propagation, they are heavily correlated in LES as shown in Fig. . One can identify several highly correlated input clusters, representing (1) yaw (ϕ), (2) atmospheric inflow (δα, α, zL-1, TI) and (3) turbine variables (CT, CQ, TSR). Note that wind speed is not included since it is approximately constant in all simulations and correlated with both inflow and turbine parameters. A high correlation between variables indicates that they contain much of the same information. Providing the same information to a model multiple times is futile as it will not improve the accuracy. For this reason, it is hypothesized that reasonable accuracy can be achieved with the regression model as long as one variable from each input cluster is included. This would reduce the number of model parameters and would give the user freedom to choose parameters based on preference and availability. However, since the input variables are not perfectly correlated, the information they contain is slightly different, and including both variables can increase the model’s accuracy. For this reason, two versions of the surrogate model having a different number of variables are experimented with; see Sect. . Although a high correlation between input variables is usually undesirable in regression problems due to multicollinearity, Sect.  explains that this is not an issue due to the regression model used in this study.

This regression model is linear, so to include nonlinear relations the input variables can be transformed using reciprocal (f(x)=x-1), exponential (f(x)=ex), logarithmic (f(x)=ln(x)) or square root (f(x)=x)) transformations. All these transformations have been tested in the procedure described in Sect. . In addition to the transformed variables, second-order polynomial and interaction terms are added, as well as an intercept (unity), extending the set of input parameters.

Since the LES data set has a relatively small sample size, a linear model is chosen as they perform well on small sample sizes, reduce the risk of overfitting compared to more complex machine learning models and are highly interpretable .

The regression is formulated as a linear model in matrix form 1Y(n×d)=X(n×p)×B(p×d), which estimates the output variable Y based on the design matrix X and coefficient matrix B. Matrix dimensions indicated in Eq. () represent the sample size n, number of downstream distances d and number of input parameters p. Note that p contains the transformed variables and their second-order and interaction terms as well as intercepts. Since these parameters are highly correlated and not all relevant, the coefficients are determined based on a lasso regression method as introduced by . This guarantees a shrinkage of the number of variables through a regularization parameter found by cross-validation. Relevant input parameters are isolated from irrelevant parameters by multiplying the latter with a coefficient of zero, effectively eliminating them from Eq. (). Multicollinearity is therefore not an issue in lasso, contrary to ordinary least squares, as typically only one parameter is chosen from a set of highly correlated input parameters. This reduces the number of input parameters, increasing the interpretability of the model. Additionally, it is desired that the same set of input parameters is used to estimate the output variable at all downstream distances. This is guaranteed in the multi-task lasso method introduced by , which is implemented in the multi-task lasso algorithm from the scikit-learn Python library . See Appendix  for further explanation.

Whereas fitting the regression coefficients is more complex than ordinary least squares fitting, the estimations of the key wake steering variables in the testing or execution phase are generated through simple matrix multiplication as shown in Eq. (). The algorithm is therefore highly interpretable, easy to implement and computational inexpensive.

The coefficient matrix B can be used to estimate the key wake steering parameters in Table  from inflow variables. This information is used to compose a vertical cross-section of the wake deficit using the reverse of the multiple 1D Gaussian method described in Sect. . The amplitude of the normalized wake deficit A^ at each height (k= 1 … K) can be computed by simply filling out the Gaussian function using Az,μz,σz. Similarly, local wake center positions μ^ and local wake widths σ^ can be found by filling out a second-order polynomial. Additional assumptions outside of the rotor area are that the curl continues (dashed red line in Fig. c), and the wake width can be described by an ellipse between lower tip and surface and between upper tip and wake top (dashed red line in Fig. d). Finally, a simple 1D Gaussian can be filled out at every vertical level using the information from A^,μ^,σ^, resulting in a two-dimensional grid filled with und values (Fig. e). Comparing this composed wake to the original LES in Fig. a, one can see that this simple description still contains much of the original information. The shape of the wake is conserved, as well as the displacement of the wake center. The maximum deficit of the composed wake center appears to be slightly larger than in LES. Additionally, in the composition the maximum wake deficit is always in the center (definition of a Gaussian), which is not necessarily true in LES or reality.

Numerous combinations of input parameters are possible. This includes choosing from the variables presented in Table , as well as which of the five transformations proposed in Sect.  to use. In order to find the most accurate solution, all combinations are tested. The combination that provides the minimum absolute percentage error in available power over all training data, i.e., all considered simulations and downstream distances (4 D ≤x≤ 10 D), is sought. When using all eight variables presented in Table , denoted DART-8, only the variable transformations need to be decided. The number of possible combinations is proportional to the number of transformations to the power of the number of variables. All five transformations are tested on all variables except for ϕ, for which the logarithmic and square root transformations have been omitted because negative values occur. This results in a total of 31⋅57=234375 possible combinations. Not only is using all variables computationally expensive, as is discussed in Sect. , operationally it is also unlikely that all variables are routinely obtained due to high costs. As hypothesized in Sect. , using one variable from each input cluster is already expected to produce accurate results. To test this, a version of DART with only three variables, denoted DART-3, is considered. Allowing each variable to be chosen and transformed, the total number of possible combinations is a multiplication of the possible combinations of each input cluster. In total, (1⋅3)⋅(4⋅5)⋅(3⋅5)=900 possible combinations are tested to find the optimal set of input parameters. It should be noted that other feature selection procedures could be considered to reduce the computational expense needed for training, but enhancing the training procedure was considered outside of the scope of the current work.

DART is benchmarked against the Gaussian (GAUS) and the Gaussian-Curl Hybrid (GCH) models present in version 2.2.2 of the FLORIS framework . Although secondary steering is not studied here, the GCH is still included because of its incorporation of initial wake deflection and the added wake recovery term. Both models share the same tuning parameters for the far-wake onset (αfloris,βfloris) and wake recovery rate (ka,floris,kb,floris). Analogous to the training of DART discussed in Sect. , the values of the tuning parameters are determined by minimizing the APE of available power over all considered simulations and downstream distances (4 D ≤x≤ 10 D). Information on inflow (e.g., uh, TI) is taken from the LES data. The models are trained independently of each other and will therefore have different values for the tuning parameters.

The data used for the tuning include simulations with yaw and pitch angles. FLORIS adjusts the thrust coefficient numerically for yaw angles, but not for pitch angles. For this reason, the thrust coefficient lookup table was adjusted by the ratio CT,pitch/CT,nopitch found in LES (Fig. ).

This section displays the performance of the Data-driven wAke steeRing surrogaTe model (DART) and the benchmark models when using all 120 simulations for training or tuning. In Sect.  and  a validation of the model with testing data is shown.

Following the feature selection procedure as described in Sect. , the optimal combination of input parameters of DART-8 was found to be the set (ϕ,δα,α-1,ln(zL-1),TI-1,CT-1,CQ,TSR-1) and for DART-3 (ϕ,δα-1,ln(CT)). Figure  compares the performance of these versions to that of the benchmark models as a function of downstream distance. The shaded areas indicate a significant improvement (green), insignificant difference (yellow) or significant decline (red) in the DART accuracy compared to the best-performing benchmark model. Statistical significance is determined using an independent Welch t test on the absolute percentage error with a probability value < 0.05. This test assumes a normal distribution but can deal with unequal variances between data sets. From Fig.  it is clear that both DART-8 and DART-3 consistently provide significantly more accurate results than GAUS and GCH. Most striking is the variability in the benchmark models that is an order of magnitude larger than that of DART. The reason for this is systematically evaluated in Sect.  and . The systematic error, indicated by the median, is however very similar for all models. Comparing the two benchmark models, it is clear that GCH consistently estimates a higher power than GAUS due to the added wake recovery term. The accuracy of DART-8 is higher than that of DART-3, especially closer to the turbine. This is attributed to the stronger wake deficit closer to the turbine as the wake center deficit Az exhibits a larger range of possible values closer to the turbine. For instance in the training data at x= 4 D, the range is -0.6 ≤Az≤ -0.21, whereas at x= 10 D the range is -0.27 ≤Az≤ -0.09. Estimations with the same relative error therefore bear a larger absolute error closer to the turbine. Having access to more information, DART-8 consistently has a smaller relative error when estimating Az than DART-3, which has a larger effect on the available power estimates closer to the turbine.

The order of magnitude of computational costs needed to train the models on a single node is displayed in Table . Computational expenses needed to generate the LES database are not considered. The benchmark models tune their parameters in approximately 7.5 h (GAUS) and 8.25 h (GCH). DART's training procedure is split up in different stages. The column “Iteration” refers to the regression fitting to obtain the coefficient matrix B (Sect. ) and the calculation of the absolute percentage error in available power at 4 D ≤x≤ 10 D (Sect. ). This can be carried out in seconds, in which fewer variables result in faster fitting. The column “Combinations” indicates the number of possible combinations that need to be tested (Sect. ). The total training time is then simply the number of combinations to be tested times the execution time of one iteration. Because of its large number of possible combinations, DART-8's total training time would be over a year on a single node, which is not operationally feasible. To generate the results in Fig. , the training process was heavily parallelized. With 900 possible combinations DART-3 can be trained in approximately 11.25 h, which is only slightly more than the benchmark models. As mentioned in Sect. , the training procedure could be enhanced, but this was considered outside of the scope of the current work. Even though Fig.  shows a small accuracy gain of DART-8 over DART-3, the computational costs to train DART-8 are much larger, and measuring all these variables in the free field is impractical. For these reasons, it is decided to only consider DART-3 in the remainder of this study.

A simple leave-one-out cross-validation technique is used to discuss the performance of DART compared to the benchmark models. The models are trained or tuned with seven out of the eight BLs (Fig. ) and tested on the remaining one, representing a new inflow condition. Eight evaluations can therefore be performed, i.e., each BL being tested once. Note that for each evaluation a set of optimal parameters and transformations are determined, which can differ from DART-3 in Fig. . Similarly, GAUS and GCH are tuned again, resulting in new values for their tuning parameters. Since the models show similar behavior in relation to the downstream distance as discussed in Sect. , here only the collective result over 4 D ≤x≤ 10 D is discussed.

Figure  presents the results of this validation procedure. For all BLs, DART-3 shows a significant improvement over GAUS and GCH. The systematic biases (indicated by the medians) are similar for all models on the order of a few percent, but the variability is greatly reduced in DART-3. The main reason for this is that the benchmark models do not include a pitch angle parameter β. Although the CT tables in the models are corrected in this study, the tunable parameters do not account for this. To clarify, LES finds a decreasing wake size (in both horizontal and vertical extent) with increasing β. This is accurately captured by DART-3, but GAUS and GCH produce a wake of similar size independent of β or CT. The inclusion of this effect is a notable improvement of DART that is important for control strategies such as axial induction control e.g.,.

Furthermore, BL5 contains the worst results for all models. Figure  indicates that this is an extreme case as it has the highest Obukhov stability parameter and veer along with the lowest turbulence intensity. This is problematic for the models since it is an inflow condition unlike anything it was trained for. This indicates a limited generalizability of all models, and caution is needed when applying them under conditions that differ greatly from those used for training. This is further discussed in Sect. .

For a fair comparison between DART-3 and the benchmark models, this section only considers simulations representing operation without derating the turbine (β= 0∘). The training (selection of parameters for DART-3) and tuning (tuning parameters of GAUS and GCH) have been repeated, and the results of the leave-one-out cross-validation technique are displayed in Fig. . The variability in the benchmark models in (near-)neutral conditions (BL 1, 2, 7 and 8) decreases considerably, but DART-3 still produces significantly more accurate results. In (weakly) stable boundary layers (BLs 3 to 6) GAUS and GCH still show a large variability and occasionally a large systematic bias, which is not true for DART-3. These results suggest that DART-3 outperforms the benchmark models, especially under stable stratifications, those conditions where wake steering is deemed most effective. Furthermore, the model performance is assessed for partial-wake operation. Figure  compares the models when the downstream turbine is moved 0.5 D to the left (from the upstream observer's point of view). Generally, the variability is greatly reduced since the deficit is smaller. The benchmark models display a systematic negative bias in all BLs, which is not true for DART-3. Only in BL8 does DART-3 not show a significant improvement over the benchmark models, but no satisfying explanation has been found as to why exactly this BL displays this behavior.

A case study is displayed in Fig. a that presents the LES wake in a weakly stable boundary layer (BL3) for a turbine with ϕ= +30∘. The wake has a clearly defined curl and a wake center left of the hub. The DART-3 wind field in Fig, b shows that the wake shape and center position are well presented. The GAUS model (Fig. c), however, produces a circular wake shape and a larger wake deflection to the left. The percentage errors indicated in the top of the figure show that DART-3 has a high accuracy for both virtual turbines, but GAUS has large biases due to the misplacement of the wake center. Under stable conditions the wind veer is relatively high, adding a crosswise force pointing towards to right above hub height. This force effectively opposes the lateral thrust force component introduced by yaw misalignment pointing to the left, reducing the deflection of the wake. The opposite is true for negative yaw angles, where wake deflection is enhanced by veer. This asymmetry has already been pointed out in , and . This effect is implicitly included in DART-3, but not in the benchmark models. Figure d illustrates that these models show an ever further deflecting wake, whereas DART-3 settles at a smaller lateral displacement close to LES. This explains not only the negative bias of the benchmark models in Fig. , but also their larger spread observed in Fig. . This result strengthens the previous indication that DART-3 is superior under stable stratifications.

Although the results presented in Sect.  are encouraging and are believed to show proof of concept, they are not directly generalizable. A data-driven surrogate model is sensitive to the data used for training, and encountering situations that vary greatly from those used for training can result in large errors. This includes very dissimilar atmospheric conditions, as already illustrated by the strongly stable BL5 in Fig. , but extends to other locations (e.g., topography, wind farm layout) and turbine types. Generating a numerical database with more atmospheric conditions, tailored to each location and turbine type, is not possible due to the high computational expense of these high-fidelity models. This limits a large-scale implementation of data-driven surrogate models trained with numerical data. Potentially, field measurements could be used, either in isolation or in combination with numerical data. Wake data could possibly be obtained from long-range lidars or strain measurements from the turbine’s blades . Exploration of these possibilities is deemed an important task for future research.

In this exploratory study, the development of DART was limited to the far wake and a two-turbine setup. If desired, further development of the model is needed to include the near wake, which can for instance be done by including the super-Gaussian description e.g.,. An extension to wind farm level could be achieved by for instance applying the superposition principle as done in GAUS and GCH, although the accuracy of DART under disturbed inflow needs attention.

As mentioned in Sect. , analytical models such as GAUS and GCH are presumed to be more robust than purely data-driven models. However, when properly trained, the accuracy of DART is expected to be significantly higher than that of analytical models as it is specifically tailored to certain scenarios. This can easily be understood by looking at the number of fitted or tuned parameters. Since DART includes second-order polynomial and interaction terms, adding more input variables exponentially increases the size of coefficient matrix B (Eq. ). This means that for DART-3, having only 3 input variables, B contains 10 coefficients, but with 8 input variables DART-8’s B already contains 45 coefficients. When comparing this to the four tuning parameters of the benchmark models, one can understand why the latter are more robust but also are expected to have a lower maximum achievable accuracy.

To demonstrate DART's interpretability, Fig.  illustrates DART-3’s fitted regression coefficients for all 10 input parameters for μy at x= 6 D. Since the order of magnitude of the input parameters can vary greatly, for this example the input parameters were scaled between -1 and 1 before regression fitting. Consequently, the fitted coefficients indicate how important each input parameter is in estimating the output variable. For the lateral wake center displacement it can easily be seen that ϕ is the dominant parameter, which intuitively makes sense. Other important parameters are the interaction term ϕ⋅ln(CT) (turbine variable cluster), y0 (intercept), δα-1 and δα-2 (atmospheric inflow cluster), while other parameters only slightly affect the wake center displacement.

Alternative to the interpretable lasso model, more complex black-box models (e.g., neural networks) could be considered as they are expected to have a higher accuracy when abundant data are available. Simpler models are, however, always preferred because they are less prone to overfitting, which is especially true for small sample sizes as used in this study. In addition, a model's interpretability typically diminishes with increasing complexity.

A simple evaluation of computational costs has been carried out to ensure that DART is sufficiently computationally efficient. The speed test comprises producing cross-sections downstream of the turbine and therefore excludes the computational resources needed to generate the LES data and to train or tune the models. This test was executed on a laptop running Ubuntu 20.04.1 with eight 1.80 GHz Intel i7-8550U CPU's and 8 GB RAM, having a minimum number of processes running in the background. All files containing relevant information, such as inflow variables, were stored locally at the same location. Run times are given as an average and standard deviation over 40 iterations, representing all simulations with β=0∘, such that no adjustment of the benchmark's thrust coefficient lookup table is needed. Table  shows that when producing results for the whole region considered in this study (4 D ≤x≤ 10 D), the run time of DART is comparable to GCH and slightly higher than GAUS. When simulating only one downstream distance, for instance exactly where a turbine is located, DART performs similarly to GAUS. These results suggest that DART is quick enough to be used for controlling purposes.